Crystal surfaces in and out of equilibrium: A modern view

The last two decades of progress in the theory of crystal surfaces in and out of equilibrium is reviewed. Various instabilities that occur during growth and sublimation, or that are caused by elasticity, electromigration, etc., are addressed. For several geometries and nonequilibrium circumstances, a systematic derivation provides various continuum nonlinear evolution equations for driven stepped (or vicinal) surfaces. The resulting equations are sometimes different from the phenomenological equations derived from symmetry arguments such as those of Kardar, Parisi, and Zhang. Some of the evolution equations are met in other nonlinear dissipative systems, while others remain unrevealed. The novel and original classes of equations are referred to as “nonstandard.” This nonstandard form suggests nontrivial dynamics, where phenomenology and symmetries, often used to infer evolution equations, fail to produce the correct form. This review focuses on step meandering and bunching, which are the two main forms of instabilities encountered on vicinal surfaces. Standard and nonstandard evolution scenarios are presented using a combination of physical arguments, symmetries, and systematic analysis. Other features, such as kinematic waves, some aspect of nucleation, and results of kinetic Monte Carlo simulations are also presented. The current state of experiments and confrontation with theories are discussed. Challenging open issues raised by recent progress, which constitute essential future lines of inquiries, are outlined.

Figure 1

(Color online) Roughening transition. (a) Facetting of ${}^4\mathrm{He}$ crystal. From top to bottom, the temperatures are successively 1.4, 1, 0.4, and $0.1\mathrm{K}$. The size of the facets is larger than on equilibrium shapes due to slow growth. From 9. (b) and (c) NaCl crystal. From 72. From (b) to (c), the temperature is decreased leading to the appearance of a facetted shape with an atomically flat surface.

Figure 2

(Color online) Screw dislocations. (a) Atomic force microscope picture of a screw dislocation during the growth of an insulin crystal (219; 66). (b) Low distortion reflection electron microscope (REM) image of a screw disclocation on Si(111) (118).

Figure 3

Schematic view of the surface during 2D growth.

Figure 4

Summary of various atomic processes on a vicinal surface. Deposition (with a flux $F$), diffusion (with $D$ as the diffusion constant), desorption (with rate $1∕\tau$), and step attachment or detachment (with rate ${\nu }_{\pm }$ from each side) is shown. $D_L$ represents the line diffusion along the step.

Figure 5

(Color online) Instabilities on vicinal surfaces. (a) Meandering instability on Si(001). From 130. (b) Bunching instability with three macrosteps formed from bunching of many monatomic steps. From 199.

Figure 6

Changes in the morphology of the Si(111) surface during the deposition of Si. The growth rate is $0.5\mathrm{\AA }∕\mathrm{s}$. (a) Before growth. (b) One monolayer deposited. (c) Two monolayers deposited. (Dark field transmission electron microscope images of a $\mathrm{Si}{\mathrm{O}}_2$ coated surface.) The surface is initially smooth and becomes rough after a meandering instability (201).

Figure 7

(Color online) Atomic steps wandering at equilibrium: (a) STM image of steps on Cu(100) vicinal surface. The fuzziness accounts for the fast fluctuations (61). (b) REM image of Si(111) steps. Courtesy of J.-J. Métois.

Figure 8

Top view of a step of meander $\zeta$. The average step tangent vector is along $x$ and the average normal is along $z$. The arclength is denoted as $s$ and $\delta \theta$ is the angle between $z$ and the normal $\mathbf{n}$.

Figure 9

A schematic view of (a) a vicinal surface inclined down in the $z$ direction. Steps run parallel to the $x$ direction on average. (b) A top view. The deviation of the $m\text{th}$ step from its regular position is ${\zeta }_m\left(x\right)$.

Figure 10

Schematic view of a vicinal surface. $D$ is the diffusion constant, $F$ is the deposition flux, $\tau$ is the desorption time, and ${\nu }_{\pm }$ are step attachment coefficients from the lower and upper sides, respectively. The potential barrier to jump over the step is denoted as $W_s$.

Figure 11

Three basic mass transport mechanisms for an isolated step: (1) edge diffusion, (2) terrace diffusion (nonlocal mechanism), and (3) attachment-detachment (local mechanism).

Figure 12

Low temperature step relaxation. (a) Atomistic picture is coarse-grained to (b) a model with a continuous step profile and a continuous concentration of atoms along the step.

Figure 13

(Color online) Formation of a shock when the velocity $Q^{\prime }$ increases with the density $\rho$, i.e., $Q^{″}>0$. The plots (1)–(3) are in chronological order.

Figure 14

Schematic view of the evolution of the bunches. Top panel: $Q^{″}>0$, the bunches are convex. Bottom panel: $Q^{″}<0$, the bunches are concave.

Figure 15

STM image and cross section of vicinal Si(111) surface etched for $5\min$ at room temperature with KOH. From 58.

Figure 16

Two schematic descriptions of the meandering instability mechanism in the presence of an Ehrlich-Schwoebel effect. (a) Point effect. The higher density of isoconcentration lines at the tip of protuberances indicates a larger attachment mass flux, leading to faster growth, thus amplifying the perturbation. (b) A strip of width ${\mathcal{L}}_c$ feeds the step. The area feeding a small step element of length $ds$ depends on the local curvature.

Figure 17

A typical time evolution of the KS equation (3.26).

Figure 18

A typical pattern obtained from a 2D numerical solution of Eq. (3.31): (a) first, ripples form from the linear meandering instability; (b) then, chaos takes place.

Figure 19

Top view of an element of teracce area between two steps along $AB{A}^{\prime }{B}^{\prime }$ and $C{C}^{\prime }$ under step meandering.

Figure 20

Time evolution of step meandering as found from the solution of highly nonlinear equation (3.44), showing that the wavelength is fixed at early stages.

Figure 21

Evolution of the meander roughness with time, corresponding to Fig. 20.

Figure 22

A typical time evolution of the meander of a single wave in Fig. 20. The step develops plateaus.

Figure 23

Elastic effects on coarsening. (a) Without elasticity, $C_1=0$ and $C_2\to \infty$ in Eq. (3.70), the wavelength is frozen. (b) With elasticity, $C_1=0.2$, endless coarsening is observed.

Figure 24

Fourfold step anisotropy of $A_{\Gamma }$ or $A_L$. Top left: A typical polar plot of an anisotropic function where $\theta =0$ is a maximum. In ${\theta }_c$, the index $c$ stands for $\Gamma$ or $L$. Bottom left: A typical picture of the step profile (where the scenario of endless growth of the amplitude and frozen wavelength still holds). Top right: The function is maximum at an angle $\theta \ne 0$. Under some conditions the step may temporarily be pinned along the maximum anisotropy direction, as shown by the dotted lines at bottom right. (This case may lead to interrupted coarsening.)

Figure 25

A snapshot of a typical pattern when allowance is made for anisotropy [Eq. (3.79)]. Interrupted coarsening may take place. Here we show the case where steps are stabilized by anisotropic line diffusion and isotropic line tension. We have chosen $A_{\Gamma }\left(\theta \right)=1$ and $A_L\left(\theta \right)=1+0.92\cos \left[4(\theta -\pi ∕4)\right]$.

Figure 26

Snapshot of a train of unstable steps. Solution of the coupled highly nonlinear evolution equations (63; 34, 35).

Figure 27

(Color online) During growth on a vicinal surface, atoms from a molecular beam land on terraces and become adatoms. Adatoms diffuse on the terrace up to a step and become mobile step atoms. Mobile step atoms diffuse along steps. If line diffusion is stabilizing, mobile step atoms diffuse toward the concave parts of the steps (as shown with arrows), thereby creating an uphill mass flux.

Figure 28

(Color online) Step meandering during MBE growth on (a) Cu(1,1,17) and (b) Cu(0,2,24). From 110.

Figure 29

(Color online) STM images of step meander on Si(111). (a) From 130. (b) From 73.

Figure 30

(Color online) Consequences of the ES effect on the stability of a train of steps. The arrows represent the step velocities, which are proportional to the downhill terrace width. The dotted line indicates the motion of the steps. For both (a) and (b), we have plotted the steps and their velocities for an initial configuration where one terrace is larger than the others. Below, the steps are shown at a later time, having subtracted step motion of the unperturbed surface for the sake of clarity. (a) During sublimation, the wider terrace becomes increasingly wider resulting in step-bunching instability. (b) During growth the opposite is obtained (see text) and the surface is stabilized.

Figure 31

The dispersion relation (4.26) for the three different cases: (i) $a_2<0$, (ii) $0<a_2<2a_4$, and (iii) $a_2>2a_4$.

Figure 32

Benney dynamics: (a) Dynamics of the step density $\rho =1∕{\partial }_{y}z$ [$z\left(y\right)$ is the surface profile] from the Benney equation (4.38) with $b=24.6$. (b) Evolution of the step density $\rho$ from the full solution of the step model for 200 steps [Eqs. (4.13, 4.14)]. Parameters are chosen such that $D=1$ and $\ell =1$. We took $\tau =0.28$, $A=0.01$, $c_{\mathrm{eq}}^0=10$, and ${\nu }_{-}=1$. These parameters lead to $b=24.6$ from Eq. (4.40).

Figure 33

Surface profile showing coarsening in a conserved system when dynamics is described by Eq. (4.41).

Figure 34

STM observation of the Si(111) vicinal surface as a function of the direction of heating current (arrow) and the temperature range. In the upper panel the current is ascending $(I<0)$ and thus the vicinal surface is stable in regimes I and III (left and right panels), while it is unstable in the middle range (regime II). Reversal of the direction of the electric current (lower panel) inverts the situation. From 218.

Figure 35

Surface morphology of Si(111) under electromigration. $I>0$ for downhill current and $\sigma$ is the supersaturation. $\sigma >0$ corresponds to growth and $\sigma <0$ corresponds to sublimation. (a) Temperature ranges I (low temperatures) and III (high temperatures). Bunching instability occurs irrespective of the supersaturation sign (no difference between sublimation and growth). (b) Intermediate temperatures, range II. The bunching instability critically depends on the sign of the supersaturation. There are also additional instabilities. The “pairs” region corresponds to the case where step pairs form, while in the “C” region small bunches are observed whose width does not increase with time. ${\tilde{F}}_p$ denotes a critical supersaturation below which step pairs form.

Figure 36

(Color online) REM image of pairs of steps on Si(111) under dc current. From 145.

Figure 37

Late-time morphology of bunches (the quite visibile macrsotep) with antibunches (less visible in between the macrosteps). Courtesy of E. D. Williams

Figure 38

(Color online) Instability of a vicinal surface under migration. The sawtoothlike profile (4.50) of the concentration is plotted. Black dots indicate the average concentration $\overline{c}$ at the steps. If $\overline{c}>c_{\mathrm{eq}}$ $(\xi >0)$, the step moves forward, and if $\overline{c}<c_{\mathrm{eq}}$ $(\xi <0)$, it moves backward. The surface is (a) destabilized for downhill migration and (b) stablilized for uphill migration. The large arrow shows the migration direction and the small ones indicate step motion.

Figure 39

A step bunching scenario for conserved systems. (a) Wavelength $\lambda$ of the periodic steady-state solutions as a function of the largest terrace width $L_0$. The solid line represents the steady state obtained by solving the continuum and semicontinuum models, as explained in Sec. 4C3, Eq. (4.53). The arrows show that, in principle, for $\lambda >{\lambda }_m$ the system will choose the ascending branch; the arrows also show the path followed by the system upon coarsening. The solid decreasing part of the curve is not essential since the system does not explore it; it is an unstable branch. (b) Time evolution of the local step density $\rho$ obtained by the numerical integration of the nonlinear equation (4.54). (c) Full evolution of step positions in conserved dynamics.

Figure 40

(Color online) Schematic of the instability for fast step kinetics (either instantaneous step kinetics or strong step transparency) during growth and in the presence of migration. Atoms are deposited on terraces where they diffuse and attach to steps. The arrows indicate the mass flow of freshly landed atoms (i.e., atoms which have not yet reached a step). Their thickness is proportional to the amplitude of the mass flow. (a) A downhill migration force produces a downhill attachment bias on each terrace. We have assumed that one terrace is wider than its neighbors (for example, due to statistical fluctuations). The fastest step is indicated, showing that the large terrace widens for a downhill force, thereby leading to amplification of the surface perturbation, which gives rise to the instability. (b) An uphill migration force produces an uphill attachment bias. Following the same reasoning, the large terrace retracts for an uphill flux and the surface is stable. The situation is reversed during sublimation.

Figure 41

(Color online) Dispersion relation for transparent steps under growth and downhill migration (or sublimation and uphill migration). In order to focus on the consequences of migration, the elastic interactions have been neglected (i.e., $A=0$). As transparency increases, the peak splits into two. For sublimation the corresponding curves are obtained by up-down symmetry $(\mathrm{Re}\left[i\omega \right]\to -\mathrm{Re}\left[i\omega \right])$. From 140.

Figure 42

(Color online) Dispersion relation for sublimation and downhill migration (Sec. 4D2). The solid curve is obtained from the curve in Fig. 41 from an up-down symmetry $(\mathrm{Re}\left[i\omega \right]\to -\mathrm{Re}\left[i\omega \right])$. The dashed curve is obtained by slightly reducing transparency; opacity favors $\varphi =\pi$ as we know from Sec. 4C3a.

Figure 43

STM images of Si(100). (a) Single-step biperiodic vicinal surface. Two successive steps fluctuate with significantly different amplitudes. The strongly fluctuating one is called $S_{\mathrm{B}}$, while the other is referred to as $S_{\mathrm{A}}$. (b) Double steps. Courtesy of M. Lagally.

Figure 44

Dispersion relation (4.85) for bunching at long wavelength $\varphi \ll 1$ during heteroepitaxial growth. (a) $A_0={\tilde{\alpha }}_0+\tilde{\mathcal{S}}$ positive: instability at long wavelength. (b) $A_0=\text{negative}$ and small: instability at finite wavelength. (c) $A_0=\text{negative}$ and large: the surface is stable.

Figure 45

(Color online) Surface mass fluxes may stabilize or destabilize the surface. There are three basic instabilities: (a) mound formation induced by an uphill mass flux, (b) step bunching induced by slope-dependent variations of the mass flux, and (c) step meandering induced by an uphill mass flux. We observe that mound formation and step bunching may be described within the frame of a 1D model, while step meandering requires a 2D description of the surface.

Figure 46

Three scenarios for coarsening. (i) No steady state above the most unstable wavelength ${\lambda }_m$. No coarsening. (ii) Perpetual coarsening. (iii) Steady states up to a value of $\lambda$ corresponding to maximum of the curve $\lambda \left(A\right)$. If ${\lambda }_m$ is larger than that value, no coarsening occurs; if it is lower, interrupted coarsening takes place.